[OskarS]

It's usually referred to as "c". I'm not sure what you're asking? It's like saying "3 is the number that follows 2", and then asking "yeah, but what really IS the number 3, man?". The answer is: it's the cardinality of the sets which can be in a bijection with the power set of the natural numbers.

If you're referring to the continuum hypothesis, that is a distinct question from "what is the cardinality of the real numbers". Also: a solved one, the contiuum hypothesis is independent of ZFC, so saying "we don't even know what the cardinality is" is still wrong.

[danharaj]

This is an extremely unusual conception of epistemology, even for mathematics. This knowledge that is contingent on axioms isn't at all convincing. At the beginning of the 20th century, this was a raging debate. It wasn't quite resolved, it just didn't quite matter to practicing mathematicians so it kind of faded into the background.

There is a massive gap between 3 and the cardinality of the continuum. 3 is directly examinable. If I count some collection of objects, my fingers say, I will immediately grasp 3.

On the other hand, the set of real numbers is a highly pathological, abstract concept. Everything we know about the reals suffers from two severe deficiencies: One, it depends on infinitary axioms which presuppose facts about the phenomenon we would like to investigate. Two, even what we can infer from such axioms is always indirect evidence. That's not surprising: All reasoning about infinity is indirect.

In certain mathematical settings it is even true that the Cauchy sequence definition of real numbers and the Dedekind cut definition do not agree. At the very least, the reals isn't even a thing: there are many reals. That you say that "we know the cardinality of the reals" because we know CH is independent of ZFC is... preposterous to say the least. "If you assume you know the cardinality of the reals, then you know the cardinality of the reals; therefore we know the cardinality of the reals" is basically what such axiomatic acrobatics boils down to. Axioms are not knowledge.

[no_identd]

Here, another interesting construction of the real numbers:

https://arxiv.org/abs/math/0405454 - The Eudoxus Real Numbers

https://ncatlab.org/nlab/show/Eudoxus+real+number

https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...

(Surprisingly enough, ncat & Wikipedia complement each other here in their explanations of it.)

[gowld]

Getting back to the OP: in a very real sense, the set of Real numbers does not exist. Therefore, it is impossible to know its size via experience. The only possible way is to derive it from chosen axioms.

Note that even your conception of 3 is reliant on axioms. How do you "know" that 3 ducks is the same 3 as 3 fingers? Only via axioms.

[danharaj]

That's sounds a little absurd. A five year old child knows that 3 fingers is the same 3 as 3 ducks. Certainly 5000 years ago people understood 3 without ever knowing what an axiom is. The peano axioms didn't create arithmetic, arithmetic created the peano axioms. S(S(S(0))) follows | | |, not the other way around.

[sykh]

The cardinality of the continuum is a cardinal number. It’s one of the alephs. It is not known which aleph it is. So it’s not known what the cardinality of the powerset of the naturals is. It’s just known that it is the same cardinal number of the reals. Basically, we have two jars of marbles that contain the same number of marbles but it’s not known how many marbles that is.

[jordigh]

The continuum hypothesis being independent just means that it's an additional rule you can add or remove from the game you are trying to play. It doesn't mean we are lacking in knowledge and that if we were to work harder we would solve this problem. We do know which aleph c is: it's aleph_1 with CH and some other aleph without CH. Just take your pick which version you like better.

It's not like we don't know which one is the true model of military combat: chess or checkers. They're just two different games with two different rule sets, and you get to pick which one you like to play more.

[sykh]

The set theory that most working mathematicians deal with is ZFC. In ZFC it is not known what cardinal the continuum is. Hence the statement that I was responding to is incorrect. The person I responded to said that they do know how many reals there are.

The cardinality of the reals is called c. It is known to the be the same as the cardinality of the power set of the naturals. It is not known, in ZFC, which aleph this is. We just know that it is the same as the size of another set.

If you want to add an axiom and say that c is aleph1 then you are free to do so. But if you don't have this axiom then you don't know which aleph it is. So in what sense can you say that you know how many reals there are? You only know it if you add an axiom that says, "It is aleph1."

If I have a jar of pennies and I know it has the same number of pennies as the number of quarters in another jar that I have, does this mean I know how many pennies are in the jar?

[OskarS]

Exactly right.

[throwaway080383]

I think the issue is with the implicit claim that we don't "know" a cardinal until we know which aleph it corresponds to.